Problem: Lucy has two star systems left to visit on her voyage, but her ship is running low on fuel. The first system, KA- $7$, is $1200$ light years $(\text{l.y.})$ away while the second system, KA- $11$, is $1700\text{ l.y.}$. Her top priority is to visit star system KA- $7$. To determine if she'll also be able to visit KA- $11$, she must find the distance between KA- $7$ and KA- $11$. If Lucy sees an angle of $52^\circ$ between KA- $7$ and KA- $11$, how far are KA- $7$ and KA- $11$ apart? Do not round during your calculations. Round your final answer to the nearest light year.
Answer: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $AB=d$. $52^\circ$ $d$ $1700\text{ l.y.}$ $1200\text{ l.y.}\,\,\,\,\,\,\,\,\,\,$ $A$ $B$ $C$ Since we are given two side lengths and the angle measure between them, we can use the law of cosines. Using the law of cosines $\begin{aligned} (AB)^2&=(AC)^2+(BC)^2-2AC\!\cdot\! BC\!\cdot\!\cos(C)\\\\ d^2&=1200^2+1700^2-2\cdot 1200\cdot 1700\cdot\cos(52^\circ) \gray{\text{Substitute}}\\\\ d&=\sqrt{1200^2+1700^2-2\cdot 1200\cdot 1700\cdot\cos(52^\circ)}\\\\ d&\approx 1348 \end{aligned}$ The answer KA- $7$ and KA- $11$ are $1348$ light years apart.